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  "body": "\n## Statement\n\nLet $V$ be a finite-dimensional vector space over $\\mathbb{R}$ or $\\mathbb{C}$, with a norm $\\|\\cdot\\|$.\nThen $V$ is a [[Banach space|0BDTTM24DP]].\n\n## Remarks\n\nProof: by [[equivalence of norms in finite dimension|03MC70ZRNR]], the norm is equivalent to the sup norm of the coordinates in a basis. Equivalent norms have the same Cauchy sequences and the same convergent sequences. With the sup norm, completeness follows from the completeness of $\\mathbb{R}$ in each coordinate.\nA consequence: each finite-dimensional subspace of a normed space is complete, and thus [[closed|0W41XG4W60]].\n\n## Explanations\n\n### L1 human\n\nIn finite dimensions, a Cauchy sequence is Cauchy in each coordinate, so each coordinate converges.\nThis works for each norm, because all norms are equivalent there.\n",
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